文摘
In this paper, we associate an undirected graph 869316303210&_mathId=si1.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=2adc43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S), the annihilating-ideal graph, to a commutative semigroup S . This graph has vertex set 869316303210&_mathId=si2.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=099e30e9674453fa156f55fcebb1dd27" title="Click to view the MathML source">A⁎(S)=A(S)∖{(0)}, where 869316303210&_mathId=si3.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=f1695d3157ea4c03cbd3f01eb595b20c" title="Click to view the MathML source">A(S) is the set of proper ideals of S with nonzero annihilator. Two distinct vertices 869316303210&_mathId=si313.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=536ec557dcedcc40b911e4741b398190" title="Click to view the MathML source">I,J∈A⁎(S) are defined to be adjacent in 869316303210&_mathId=si1.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=2adc43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S) if and only if 869316303210&_mathId=si5.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=0a2d4dfc0f93db2e8d7f321671f97f0f" title="Click to view the MathML source">IJ=(0), the zero ideal. Conditions are given to ensure a finite graph. Semigroups for which each nonzero, proper ideal of S is an element of 869316303210&_mathId=si6.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=3971bbfd039685788f340bb546d3ce89" title="Click to view the MathML source">A⁎(S) are characterized. Connections are drawn between 869316303210&_mathId=si1.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=2adc43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S) and 869316303210&_mathId=si117.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=033e8fc213e87201afc6f20cf84826d8" title="Click to view the MathML source">Γ(S), the well-known zero-divisor graph, and the connectivity, diameter, and girth of 869316303210&_mathId=si1.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=2adc43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S) are described. Semigroups S for which 869316303210&_mathId=si1.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=2adc43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S) is a complete or star graph are characterized. Finally, it is proven that the chromatic number is equal to the clique number of the annihilating ideal graph for each reduced semigroup and null semigroup. Upper and lower bounds for 869316303210&_mathId=si395.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=7e15108d72d68750bad5c05f34bee6fc" title="Click to view the MathML source">χ(AG(S)) are given for a general commutative semigroup.